Introduction to Mathematical Analysis I - Second Edition

72 3.2 LIMIT THEOREMS Infinite limits of functions have similar properties to those of sequences from Chapter 2 (see Definition 2.3.2 and Theorem 2.3.6 ) . Example 3.2.7 We show that lim x → 1 1 ( x − 1 ) 2 = ∞ directly from Definition 3.2.5 . Let M ∈ R . We want to find δ > 0 that will guarantee 1 ( x − 1 ) 2 > M whenever 0 < | x − 1 | < δ . As in the case of finite limits, we work backwards from 1 ( x − 1 ) 2 > M to an inequality for | x − 1 | . To simplify calculations, note that | M | + 1 > M . Next note that 1 ( x − 1 ) 2 > | M | + 1, is equivalent to q 1 | M | + 1 > | x − 1 | . Now, choose δ such that 0 < δ < q 1 | M | + 1 . Then, if 0 < | x − 1 | < δ we have 1 ( x − 1 ) 2 > 1 δ 2 > 1 1 | M | + 1 = | M | + 1 > M , as desired. Definition 3.2.6 (limits at infinity) Let f : D → R , where D is not bounded above. We write lim x → ∞ f ( x ) = ` if for every ε > 0, there exists c ∈ R such that | f ( x ) − ` | < ε for all x > c , x ∈ D . Let f : D → R , where D is not bounded below. We write lim x →− ∞ f ( x ) = ` if for every ε > 0, there exists c ∈ R such that | f ( x ) − ` | < ε for all x < c , x ∈ D . We can also define lim x → ∞ f ( x ) = ± ∞ and lim x →− ∞ f ( x ) = ± ∞ in a similar way. Example 3.2.8 We prove from the definition that lim x →− ∞ 3 x 2 + x 2 x 2 + 1 = 3 2 . The approach is similar to that for sequences, with the difference that x need not be an integer. Let ε > 0. We want to identify c so that 3 x 2 + x 2 x 2 + 1 − 3 2 < ε , (3.5) for all x < c .